The geometry of quantum codes

Abstract

Quantum particles are continuously interacting with the environment hence quantum information is always susceptible to errors. Consequently when encoding information into quantum bits a special treatment is required such that there is a recovery map between the information sent and the information received capable to correct certain class of errors. We allow the quantum bits to take two orthogonal values (qubits) and we encode k logical qubits on n physical qubits. We first present the already well known class of quantum codes called stabiliser codes and its geometry from which one can deduce the code parameters. Finally we shall study the much less known class of quantum codes called non additive codes, result of direct sums of stabilizer codes, and for which we provide a geometric framework which appears to be new.